# number theory – Why does the proof of Theorem 3.10 in Apostol (1976) use generalized convolutions from previous chapter when there could be a simpler proof?

The proof of Theorem 3.10 in Introduction to Analytic Number Theory by Apostol goes like this:

Theorem 3.10 If $$h = f * g$$, let
$$H(x) = sum_{n le x} h(n), F(x) = sum_{n le x} f(n), text{ and } G(x) = sum_{n le x} g(n).$$
Then we have
$$H(x) = sum_{n le x} f(n) G left( frac{x}{n} right) = sum_{n le x} g(n) F left( frac{x}{n} right).$$

PROOF. We make use of the associative law (Theorem 2.21) which relates the operations $$circ$$ and $$*$$. Let
$$U(x) = begin{cases} 0 & text{ if } 0 < x < 1, \ 1 & text{ if } x ge 1. end{cases}$$
Then $$F = f circ U$$, $$G = g circ U$$, and we have
$$f circ G = f circ (g circ U) = (f * g) circ U = H,$$
$$g circ F = g circ (f circ U) = (g * f) circ U = H.$$
This completes the proof.

I am curious to understand why use generalized convolution and Dirichlet multiplication here at all.

Since section 3.5, the book has been relying on the following reordering of summation at various places,
$$sum_{n le x} sum_{d | n} f(d) = sum_{substack{q, d \ qd le x}} f(d) = sum_{d le x} sum_{q le frac{x}{d}} f(q).$$
Can we not arrive at a simpler proof with a similar reordering of summation like this:
begin{align*} H(x) & = sum_{n le x} h(n) = sum_{n le x} (f * g)(n) = sum_{n le x} sum_{d | n} f(d) gleft(frac{n}{d}right) = sum_{substack{q, d \ qd le x}} f(d) gleft(qright) \ & = sum_{d le x} sum_{q le frac{x}{d}} f(d) g(q) = sum_{d le x} f(d) sum_{q le frac{x}{d}} g(q) = sum_{n le x} f(n) sum_{q le frac{x}{n}} g(q) = sum_{n le x} f(n) Gleft(frac{x}{n}right). end{align*}
I am trying to understand if a simpler proof using basic arithmetic and reordering of summations is correct or if I am missing something and there is a specific reason why Apostol went for a proof using generalized convolution and Dirichlet multiplication.

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